Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Invertible Euclidean linear maps
Definition
Let . A linear map (A linear map in Euclidean coordinates) is invertible when there is a linear map such that
The map is unique: if has the same two properties, then . It is denoted .
Depends on
Used by
- The real complex-squaring map is locally but not globally invertible off the origin Counterexample
- The regular locus of a square-dimensional C¹ map Definition
- FALSE: an everywhere-invertible derivative gives a global inverse False statement
- Choice-free smooth inverse function theorem in Euclidean space Lemma
- Newton maps are uniform contractions near a point with invertible derivative Lemma
- An injective C¹ map with invertible derivative sends compact Jordan sets to compact Jordan sets Theorem
- The Euclidean implicit function theorem with derivative formula Theorem
- The Euclidean inverse function theorem Theorem
Dependency tree · two levels
2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. Lebl, Basic Analysis II, §8.2 Matrices and linear mappings (standard reference, not scraped)